Retrieving a singular pole from the quotient of the coefficients of the power series

Question

I want to show the following statement: Let $f$ be a meromorphic function on $\mathbb{C}$ with one and only one pole at $z_0\neq 0$. This pole can have any order $k>0$, but has to be the only pole of $f$. Let $f(z)=\sum^\infty_{n=0}a_nz^n$ be the power series representation around $0$, hence its radius of convergence has to be $|z_0|$ since that is the first and only singularity of $f$. To show: $$\lim_{n\to\infty}\frac{a_n}{a_{n+1}}=z_0.$$ Now, the way I think this can be shown is as follows:

1. Show the statement for $z_0=1$ and $k=1$.
2. Generalise to $k>0$.
3. Generalise to $z_0\neq 0$.

Item 1 and 3 have been shown, the first by showing (via residue theorem) that $\lim_{n\to\infty}a_n=g(1)\neq 0$ where $g$ holomorphic such that $f(z)=\frac{g(z)}{z-1}$, the second by considering the function $h(z)=f(z_0z)$, which will have one and only one pole in $1$. The problem is item 2. I have tried plenty of things but can't manage to prove the statement. It really cries for induction.

• First try: Consider the primitive $F(z)=\sum^\infty_{n=1}\frac{a_{n-1}}{n}z^n$, which will also have a radius of convergence of $1$ (since $z_0=1$). I now hope that $F$ could be extended to a primitive everywhere since then $F$ would have a pole of order $k-1$ and hence the quotient of coeffiecients would converge to $1$. Yet I can't manage to find a primitive everywhere with a singularity at $1$ which would coincide with $F$ on the unit disk.
• Second try: Multiply $f$ with $(z-1)$, then I obtain a new function with a pole of order $k-1$, but the coefficients are $a_n-a_{n-1}$ and the quotient is a quotient of differences. This I can't really modify to obtain the quotient of $a_n$.
• Third try: No induction, but instead computing the Cauchy product of the series $$\frac{1}{(1-z)^k}=\sum_{n=0}^\infty \binom{n+k-1}{k-1}z^n$$ and $g(z)=\sum_{n=1}^\infty b_nz^n$ where $g$ is holomorphic in $1$ and $g(1)\neq 0$, but this will only yield an incredibly jumbled representation of $a_n$ and I can't seem to use the fact that $g(1)\neq 0$.

Any ideas, what else I could try? Or whether one of these tries might actually yield a solution? Cheers for any help!